Optimal quantum state reconstruction for cold trapped ions

We study the physical implementation of an optimal tomographic reconstruction scheme for the case of determining the state of a multi-qubit system, where trapped ions are used for defining qubits. The protocol is based on the use of mutually unbiased measurements and on the physical information described in H. H\"{a}ffner \emph{et. al} [Nature \textbf{438}, 643-646 (2005)]. We introduce the concept of physical complexity for different types of unbiased measurements and analyze their generation in terms of one and two qubit gates for trapped ions.